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1、定义
陀螺寻北仪又称陀螺罗盘
2、真北方位
地理坐标系对应的北。
N 表示真北方位，E 表示正东方向。

nnn 系表示导航坐标系，LLL 表示地理纬度，wiew_{ie}wie​ 表示地球自转角速度。
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        <meta property="og:description" content="一、寻北仪
1、定义
陀螺寻北仪又称陀螺罗盘
2、真北方位
地理坐标系对应的北。
N 表示真北方位，E 表示正东方向。

nnn 系表示导航坐标系，LLL 表示地理纬度，wiew_{ie}wie​ 表示地球自转角速度。
3、单轴陀螺仪敏感轴...">
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1、定义
陀螺寻北仪又称陀螺罗盘
2、真北方位
地理坐标系对应的北。
N 表示真北方位，E 表示正东方向。

nnn 系表示导航坐标系，LLL 表示地理纬度，wiew_{ie}wie​ 表示地球自转角速度。
3、单轴陀螺仪敏感轴...">
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                                <h2>
                                    4.2 寻北仪 地平仪
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                                    2023-11-12, 700 words, 3 min read
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                                        <h2 id="一-寻北仪">一、寻北仪</h2>
<h4 id="1-定义">1、定义</h4>
<p>陀螺寻北仪又称陀螺罗盘</p>
<h4 id="2-真北方位">2、真北方位</h4>
<p>地理坐标系对应的北。</p>
<p>N 表示真北方位，E 表示正东方向。</p>
<img src="http://cos.pansis.site/202311092227371.png/abc123" alt="image-20231109222727285" style="zoom:53%;" />
<p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">n</span></span></span></span> 系表示导航坐标系，<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault">L</span></span></span></span> 表示地理纬度，<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>w</mi><mrow><mi>i</mi><mi>e</mi></mrow></msub></mrow><annotation encoding="application/x-tex">w_{ie}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 表示地球自转角速度。</p>
<h4 id="3-单轴陀螺仪敏感轴的输入量">3、单轴陀螺仪敏感轴的输入量</h4>
<p>1、地球自转角速度 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>w</mi><mrow><mi>i</mi><mi>e</mi></mrow></msub></mrow><annotation encoding="application/x-tex">w_{ie}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 分解</p>
<p>地球自转角速度可分解为沿当地垂线和在当地水平面内的两个矢量。</p>
<img src="http://cos.pansis.site/202311092229230.png/abc123" alt="image-20231109222933167" style="zoom:33%;" />
<img src="http://cos.pansis.site/202311092229218.png/abc123" alt="image-20231109222920169" style="zoom:53%;" />
<p>2、单轴陀螺仪敏感轴的输入量</p>
<img src="http://cos.pansis.site/202311092236135.png/abc123" alt="image-20231109223654081" style="zoom:40%;" />
<img src="http://cos.pansis.site/202311092236884.png/abc123" alt="image-20231109223634855" style="zoom:53%;" />
<h4 id="4-寻北极原理">4、寻北极原理</h4>
<p>令一个单轴陀螺的敏感轴在水平面上转动。当陀螺输出值最大时，其敏感轴指向即为真北方位。</p>
<img src="http://cos.pansis.site/202311092237476.png/abc123" alt="image-20231109223744399" style="zoom:33%;" />
<p>开始寻北时，陀螺处于位置1。假设陀螺敏感轴的初始方向与真北方向的夹角为 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.0037em;">α</span></span></span></span>。</p>
<p>陀螺仪输出角速率球自转角速率 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>w</mi><mrow><mi>i</mi><mi>e</mi></mrow></msub></mrow><annotation encoding="application/x-tex">w_{ie}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 的关系式 <img src="http://cos.pansis.site/202311092239981.png/abc123" alt="image-20231109223950920" style="zoom:50%;" /></p>
<p>陀螺仪敏感轴为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>y</mi><mi>b</mi></msub></mrow><annotation encoding="application/x-tex">y_b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>轴，因此陀螺仪输出值为<img src="http://cos.pansis.site/202311092240322.png/abc123" alt="image-20231109224025278" style="zoom:33%;" /></p>
<p>$\varepsilon(t_1) $表示在位置1时刻的陀螺仪漂移。</p>
<p>同理，在其他位置测得的角速度分别为</p>
<img src="http://cos.pansis.site/202311092241302.png/abc123" alt="image-20231109224123252" style="zoom:33%;" />
<p>令陀螺仪漂移为常量，角速度方程组为</p>
<img src="http://cos.pansis.site/202311092241544.png/abc123" alt="image-20231109224141507" style="zoom:33%;" />
<p>陀螺仪敏感轴与真北方向的夹角 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.0037em;">α</span></span></span></span>  <img src="http://cos.pansis.site/202311092241915.png/abc123" alt="image-20231109224152876" style="zoom:33%;" /></p>
<p>若测量地点的纬度已知，则只需测量1、3或2、4两个位置即可</p>
<p><img src="http://cos.pansis.site/202311092242840.png/abc123" alt="image-20231109224223812" style="zoom:33%;" />     <img src="http://cos.pansis.site/202311092242454.png/abc123" alt="image-20231109224237415" style="zoom:33%;" /></p>
<h4 id="5-公式">5、公式</h4>
<h2 id="二-地平仪">二、地平仪</h2>
<h4 id="1-垂线基准的原理">1、垂线基准的原理</h4>
<p><strong>机械摆</strong>和<strong>二自由度陀螺仪</strong>的结合。</p>
<p>摆对陀螺转子轴指示方向实施控制，在运动的载体上建立起一个稳定的垂线基准</p>
<img src="http://cos.pansis.site/202311092244176.png/abc123" alt="image-20231109224409113" style="zoom:33%;" />
<p>力矩器通过对陀螺仪施加力矩，使陀螺仪转子轴与地垂线重合。</p>
<h4 id="2-获得载体的水平姿态角原理">2、获得载体的水平姿态角原理</h4>
<p>通过测量陀螺仪转子轴方向在载体系中的三个分量，便可计算得到俯仰角和滚转角。</p>
<p>重力在地理坐标系n中的表示为    <img src="http://cos.pansis.site/202311092245483.png/abc123" alt="image-20231109224502457" style="zoom:33%;" /></p>
<p>假设绕z轴偏航 (<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>ψ</mi></mrow><annotation encoding="application/x-tex">\psi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">ψ</span></span></span></span>)、绕x轴俯仰 ( <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span></span></span></span> )、绕y轴滚转 ($\gamma $)</p>
<p>载体坐标系b相对地理坐标系n的坐标转换矩阵为</p>
<img src="http://cos.pansis.site/202311092245973.png/abc123" alt="image-20231109224522906" style="zoom:30%;" />
<p>重力在载体坐标系b的表示为</p>
<img src="http://cos.pansis.site/202311092248356.png/abc123" alt="image-20231109224853314" style="zoom:33%;" />
<p>俯仰角公式为<img src="http://cos.pansis.site/202311092249338.png/abc123" alt="image-20231109224923310" style="zoom:33%;" /></p>
<p>1、真北方位定义<br>
2、地球自转角速度 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>w</mi><mrow><mi>i</mi><mi>e</mi></mrow></msub></mrow><annotation encoding="application/x-tex">w_{ie}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 分解至地理坐标系的分量<br>
3、寻北极原理、相关公式<br>
（设陀螺敏感轴的初始方向与真北方向的夹角为 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.0037em;">α</span></span></span></span>，陀螺仪输出值为xxx）<br>
（测得四个方位的角速度，如何求 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.0037em;">α</span></span></span></span>）<br>
4、姿态角中绕z轴----、绕x轴-----、绕y轴----<br>
姿态角转坐标转换矩阵（载体坐标系b相对地理坐标系n）<br>
5、地平仪获得载体的水平姿态角原理</p>
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